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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p><dfn class="terminology">The Nonhomogeneous Equations</dfn></p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_4.html ./knowl/eq4_5.html ./knowl/eq4_4.html ./knowl/eq4_4.html">
\begin{equation}
L[y]=g(x).\tag{4.1.5}
\end{equation}
</div>
<p class="continuation">If <span class="process-math">\(Y_1\)</span> and <span class="process-math">\(Y_2\)</span> are two solutions of (<a href="" class="xref" data-knowl="./knowl/eq4_4.html" title="Equation 4.1.5">(4.1.5)</a>). Then <span class="process-math">\(Y_1-Y_2\)</span> is a solution of</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_4.html ./knowl/eq4_5.html ./knowl/eq4_4.html ./knowl/eq4_4.html">
\begin{equation}
L[y]=0.\tag{4.1.6}
\end{equation}
</div>
<p class="continuation">If <span class="process-math">\(y_1, y_2, \cdots, y_n\)</span> are a fundamental set of solutions of (<a href="" class="xref" data-knowl="./knowl/eq4_5.html" title="Equation 4.1.6">(4.1.6)</a>), and <span class="process-math">\(Y(x)\)</span> is a particular solution of (<a href="" class="xref" data-knowl="./knowl/eq4_4.html" title="Equation 4.1.5">(4.1.5)</a>), then</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq4_4.html ./knowl/eq4_5.html ./knowl/eq4_4.html ./knowl/eq4_4.html">
\begin{equation*}
y=C_1 y_1+C_2 y_2+\cdots+C_n y_n+Y(x)
\end{equation*}
</div>
<p class="continuation">is the general solution to (<a href="" class="xref" data-knowl="./knowl/eq4_4.html" title="Equation 4.1.5">(4.1.5)</a>).</p>
<span class="incontext"><a href="sec4_1.html#p-154" class="internal">in-context</a></span>
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